Paraxial Ray Tracing Research Articles

Inverse problem in seismic imaging

AbstractWe address the problem of estimating sound speeds (seismic velocities) inside the earth which is necessary for obtaining seismic images in regular Cartesian coordinates. The main goals are to develop algorithms to convert time migration velocities to true seismic velocities, and to convert time‐migrated images to depth images in regular Cartesian coordinates.Our main results are three‐fold. First, we establish a theoretical relation between the seismic velocities and the time migration velocities using the paraxial ray tracing theory. Second, we formulate an appropriate inverse problem describing the relation between time migration velocities and depth velocities and show that this problem is mathematically ill‐posed, i.e., unstable to small perturbations. Third, we develop numerical algorithms to solve regularized versions of these equations which can be used to recover smoothed velocity variations. Our algorithms consist of efficient time‐to‐depth conversion algorithms based on Dijkstra‐like Fast Marching Methods, as well as level set and ray tracing algorithms for transforming Dix velocities into seismic velocities. Our algorithms are applied to both two‐dimensional and three‐dimensional problems and we test them on a collection of both synthetic examples and field data. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Approximate ray tracing for qP-waves in inhomogeneous layered media with weak structural anisotropy

We present approximate equations for qP-wave ray tracing and paraxial ray tracing in inhomogeneous layered weakly transversely isotropic (TI) media. Inside layers, the symmetry axis direction of the TI medium is allowed to vary continuously. Approximate equations are based on perturbation theory in which deviations of anisotropy from isotropy are considered to be first-order quantities. For qP-waves propagating in a TI medium, the approximate ray-tracing and paraxial ray-tracing equations depend on three parameters and two angles defining the direction of the symmetry axis. We also present the boundary conditions at interfaces for first-order rays and paraxial rays and compute reflection/transmission coefficients to the first order. All the quantities required for evaluation of the Green’s function are calculated to the first order, except the traveltime that is calculated to the second order. The accuracy of the presented algorithm is verified on simple models with respect to exact ray-tracing results. For anisotropy of about [Formula: see text], considered in the examples presented, the relative errors in traveltime and amplitude are less than [Formula: see text] and [Formula: see text], respectively. We then consider media where the symmetry axis is conformable with the structure, named structural transverse isotropy (STI). The method is applied to simple STI models and to a realistic overthrust model, showing significant differences with vertical transverse isotropy (VTI) models not only for traveltimes, but also for amplitudes.

Calculating the power of toric phakic intraocular lenses

A toric phakic intraocular lens (IOL) implanted in the anterior or posterior chamber of the eye has the potential to correct high or excessive ametropia and astigmatism with high predictability of the postoperative refraction and preservation of phakic accommodation. The calculation of spherical phakic lenses has been described previously, but a formalism for estimating the power of toric phakic lenses has not yet been published. The purpose of this study is to describe a mathematical strategy for calculating toric phakic IOLs. The method presented in this paper is based on vergence transformation in the paraxial Gaussian space. Parameters used for the calculations are the spherocylindrical spectacle refraction before implantation, corneal power (sphere and astigmatism) and (spherocylindrical) target refraction, together with the vertex distance and the predicted position of the phakic IOL. The lens power is determined as the difference in vergences between the spectacle-corrected eye and the uncorrected eye at the reference plane of the predicted lens position. The axes of the preoperative refraction, the target refraction and the corneal astigmatism are at random (not necessarily aligned). The method was applied to two clinical examples. In example 1 we calculate the power of a phakic lens for the simple case, when the target refraction is plano and the axis of the preoperative refraction is aligned to the axis of the corneal astigmatism. In example 2, the cylindrical axis of the preoperative refraction is not aligned to the corneal astigmatism and the target refraction is spherocylindrical (and the axis of the target refraction is not aligned to the preoperative refractive cylinder or the corneal astigmatism). The calculations for both examples are described step-by-step and illustrated in a table. The calculation scheme can be generalized to an unlimited number of crossed cylinders in the optical pathway. Based on paraxial raytracing, the spherical and cylindrical power as well as the orientation of the cylinder are determined from the preoperative refraction (including vertex distance), the corneal power, the intended target refraction (including vertex distance) and the predicted position of the phakic lens implant provided by the lens manufacturer. This calculation scheme can be easily implemented in a simple computer program (i.e. in Microsoft excel or matlab).

Seismic velocity estimation from time migration

We address the problem of estimating seismic velocities inside the Earth which is necessary for obtaining seismic images in regular Cartesian coordinates. The main goals are to develop algorithms to convert time-migration velocities to true seismic velocities, and to convert time-migrated images to depth images in regular Cartesian coordinates. Our main results are three-fold. First, we establish a theoretical relation between the true seismic velocities and the ‘time-migration velocities’ using the paraxial ray tracing. Second, we formulate an appropriate inverse problem describing the relation between time-migration velocities and depth velocities, and show that this problem is mathematically ill posed, i.e., unstable to small perturbations. Third, we develop numerical algorithms to solve regularized versions of these equations which can be used to recover smoothed velocity variations. Our algorithms consist of efficient time-to-depth conversion algorithms, based on Dijkstra-like fast marching methods, as well as level set and ray tracing algorithms for transforming Dix velocities into seismic velocities. Our algorithms are applied to both two-dimensional and three-dimensional problems, and we test them on a collection of both synthetic examples and field data.

유한 물체거리를 갖는 코마수차가 보정된 2 반사경계의 곡률선형방정식

원리적인 다양한 장점에도 불구하고 취급이 복잡하고 고차 연립방정식으로 표현되는 현실적 제한으로 인하여 실제 설계과정에서 잘 적용하지 않는 자이델 3차 수차를 이용하여 구한 코마수차 계수식으로부터 근사적인 제로조건을 만족하는 유한 물체거 리를 갖는 코마수차가 제거된 2반사경계의 초기설계에 유용한 곡률선형방정식을 유도하고 그 특징을 조사한다. 즉 주경과 부경의 곡률, 주경과 부경사이의 거리, 유효초점거리로 표현된 변형된 코마수차계수로부터 코마수차계수가 제거되는 조건에서 설계변수를 구하기 위해 전산수치해석 후 나온 데이터를 기반으로 주경과 부경사이의 선형관계가 나타나는 곡률선형방정식을 구하는 것이다. 이는 유한 물점의 코마수차가 보정된 2 반사경계에서 약간의 대수적인 계산만으로 최적화의 초기 입력 데이터를 손쉽게 구할 수 있는 것을 의미한다. We derived analytically the generalized curvature linear equation useful in the initial optical design of a two-mirror system with finite object distance, including an infinite object distance from paraxial ray tracing and Seidel third order aberration theory for coma coefficient. These aberration coefficients for finite object distance were described by the curvature, the inter-mirror distance, and the effective focal length. The analytical equations were solved by using a computer with a numerical analysis method. Two useful linear relationships, determined by the generalized curvature linear equations relating the curvatures of the two mirrors, for the cancellation of each aberration were shown in the numerical solutions satisfying the nearly zero condition (<TEX>$<10^{-10}$</TEX>) for each aberration coefficient. These equations can be utilized easily and efficiently at the step of initial optical design of a two-mirror system with finite object distance.

Comparing two new camera calibration methods with traditional pinhole calibrations

Two novel camera calibration methods are compared with traditional pinhole calibration: one new method uses an analytic geometrical version of Snell's law (denoted as the Snell model); the other uses 6x6 matrix-based paraxial ray-tracing (referred to as the paraxial model). Pinhole model uses a perspective projection approximation to give a single lumped result for the multiple optical elements in a camera system. It is mathematically simple, but suffers from accuracy limitations since it does not consider the lens system. The Snell model is mathematically the most complex but potentially has the highest levels of accuracy for the widest range of conditions. The paraxial model has the merit of offering analytical equations for calibration.

Método híbrido de migração pré-empilhamento em profundidade no domínio da freqüência em duas etapas com interpolação

This paper shows the application of Phase Shift Plus Interpolation Split-Step (PSPI-SS) wavefield extrapolation operator coupled with the excitation-time imaging condition to perform prestack seismic depth migration of common shot profiles. The migration methods which are characterized by the use of different propagators for the source and receiver wavefields are called hybrid migration methods. Normally, the source wavefield is determined by the computation of the traveltimes from the solution of the eikonal equation. Here the traveltimes were calculated based on a finite difference solution of the eikonal equation (TGRID), a simple geometrical approach (FAST) and also by the paraxial ray tracing method (PARAX). For the extrapolation of the receiver data we applied the PSPI Split-Step operator, that is derived from the conventional Split-Step wave equation method using multiple references velocity. As the source traveltime computation is based on the ray assintotic theory the velocity field needs to be smoothed and the migration results obtained with the Kirchhof migration method, that is a very fast migration procedure, was used to choose the best level of smoothing of the velocity field. In this work the synthetic Marmousi data and a real dataset from Gulf of Mexico were used to test the migration methods proposed here. The migration results with the hybrid methods shown better images in quality then the kirchhoff's results and comparable to those obtained with the full PSPI-SS migration method, but a low computation cost.

ABCD transfer matrices and paraxial ray tracing for elliptic and hyperbolic lenses and mirrors

A method is introduced for evaluating the ABCD transfer matrices for paraxial rays refracted/reflected by lenses/mirrors with elliptic or hyperbolic surfaces, which relies on the use of certain cusp points of the evolutes of the curves. Paraxial ray tracing techniques are developed. Applications examined include early hypothetical refracting telescopes by Descartes and Gregory.

Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis

Matrix-based Gaussian paraxial ray-tracing is commonly used for first estimates in early design stages of optical systems. However, the usual techniques are suitable only for rays in systems with a common straight-line optical axis, and cannot be used in systems containing mirrors and/or prisms. Attard extended these methods to matrix-based skew ray-tracing for systems containing cylindrical lenses with orthogonal cylinder axes and spherical lenses with combinations of cylindrical lenses, but had problems with non-straight and non-coplanar optical relations. This paper develops a novel general matrix method for paraxial skew ray-tracing in systems with non-coplanar optical axes containing spherical and flat boundary surfaces. First-order Taylor series expansion is used to approximate skew ray-tracing equations in simple repetitive linear matrix form. Sufficiently good accuracy is obtained if the proposed method is restricted to skew rays in the immediate neighborhood of the optical axis, as demonstrated by numerical examples. This study extends earlier Matrix-based paraxial ray-tracing design technique to include non-coplanar systems containing mirrors and/or reflecting prisms such as projectors, with special application potential for first estimates in early design stages of 3D optical systems.

Fresnel volume migration of multicomponent data

If the aperture of a seismic reflection experiment is strongly limited, Kirchhoff migration suffers from strong artifacts attributable to incomplete summation. This can be overcome by restricting the migration operator to the region that physically contributes to a reflection event. Examples of such limited-aperture experiments include data acquisition in boreholes, tunnels, and mines. We present an extension to three-component (3C) Kirchhoff prestack depth migration, where the migration operator is restricted to the Fresnel volume of the specular reflected raypath. We use the measured polarization direction at a 3C receiver to determine points of specular reflection. In homogeneous media, the polarization angle of 3C data can be used directly to decide whether a certain image point belongs to the Fresnel volume of a specular reflection. In heterogeneous media, the Fresnel volume around an image point is approximated by means of paraxial ray tracing. The method is tested on a synthetic vertical seismic profiling experiment with strongly limited aperture. Migration artifacts and crosstalk effects from converted waves are strongly reduced compared with standard migration schemes. The method is successfully applied to seismic data acquired in a tunnel.

유한 물체 거리를 갖는 2 반사경계의 곡률 선형 방정식

본 연구는, 원리적인 다양한 장점에도 불구하고 현실적인 제약으로 인해 실제 설계과정에서 잘 적용되지 않는, 자이델 3차 수차를 간편하게 다룰 수 있는 방안을 제안한다. 먼저 유한 물체거리를 갖는 2 반사경계에 대해 자이델 3차의 구면수차계수를 유도한다. 여기서, 유도된 구면수차계수는 고차의 비선형 방정식으로 표현되는데, 그 구성은 설계변수(물체거리, 주경 및 부경의 곡률, 주경과 부경 사이의 거리)와 유효초점거리로 이루어진다. 해석적으로 표현된 고차의 비선형 구면수차 방정식은 컴퓨터를 이용한 수치기법에 의해 근사적인 제로조건을 만족하도록 풀려진다. 이렇게 구해진 다양한 수치 해들을 주의 깊게 통찰하면 주경과 부경의 곡률 간에 선형성이 존재함을 파악할 수 있다. 즉, 결과적으로 주경과 부경의 곡률들을 선형맞춤(linear fitting)하면 곡률선형방정식이 얻어지는데, 이의 의미는 약간의 대수적인 계산으로 최적화의 초기 입력 데이터를 손쉽게 얻을 수 있는 가능성을 제시한 것이다. 한편, 응용외의 순수 수차론적인 관점에서 본다면, 본 연구의 특징은 유한 물체거리를 갖는 2 반사경계의 주경 및 부경의 곡률들이 구면수차가 거의 제로가 되는 조건 하에서 상호간에 선형 관계가 존재하였다는 것이다. In this paper, we propose easily tooling method for Seidel third order aberration, which are not well utilized in actual design process due to the complication of mathematical operation and the difficulty of understanding Seidel third order aberration theory, even though most insightful and systematic means in pre-designing for the initial data of optimization. First, using paraxial ray tracing and Seidel third order aberration theory, spherical aberration coefficient is derived for a two-mirror system with a finite object distance. The coefficient, which is expressed as a higher-order nonlinear equation, consists of design parameters(object distance, two curvatures, and inter-mirror distance) and effective focal length(EFL). Then, the expressed analytical equation is solved by using a computer with numerical analysis method. From the obtained numerical solutions satisfying the nearly zero coefficient condition(<TEX>$<10^{-6}$</TEX>), linear fitting process offers a linear relationship called the curvature linear equation between two mirrors. Consequently, this linear equation has two worthy meanings: the equation gives a possibility to obtain initial design data for optimization easily. And the equation shows linear relationship to a two-mirror system with a finite object distance under the condition of corrected third order spherical aberration.

First-order ray tracing forqSwaves in inhomogeneous weakly anisotropic media

SUMMARY First-order ray tracing equations and paraxial ray tracing system are derived for quasi-shear waves propagating in a smooth inhomogeneous weakly anisotropic medium. The first-order equations are based on explicit formulae in terms of slowness vector and elastic parameters. They only depend on 15 independent combinations of elastic parameters. For isotropic media, they reduce to exact equations. The behaviour of the approximate equations is studied in the vicinity of singularities. Ray tracing equations behave properly if the wave mode is changed when the ray crosses the conical or the intersection singularity, but the paraxial ray tracing system may be singular. A second-order correction of the traveltime is obtained by integration along the approximate rays. Comparison of the exact and approximate traveltimes in transversely isotropic and orthorhombic homogeneous media shows small relative errors except in the close vicinity of cusps.

Adjustable generation of bottle and hollow beams using an axicon

The focusing properties of a Bessel beam generated using an axicon are investigated. The distance between the axicon and the focused lens is tuned to transform the focused beams from bottle beams, with a dark focus surrounded by regions of higher intensity, to hollow beams. The evolutions of these beams are illustrated simply using geometrical structures and the paraxial ray tracing, and are experimentally verified. A specific case in which the focusing behavior approaches that of an ideal Bessel beams is also considered; the experimental results agree with theoretical predictions.

Paraxial ray tracing for atmospheric wave propagation

Ray tracing for sonic wave propagation in a two‐dimensional atmosphere structure is performed in the presence of spatially variable wind by using an Hamiltonian approach. Paraxial ray tracing is deduced by first‐order perturbations. Using standard atmospheric data or illustrative models, we test numerically the Hamiltonian approach. The ray tracing allows accurate estimation of trajectories and associated travel‐times as well as stable and accurate amplitude variations along a ray. Lateral variations of both sonic and wind velocity are important for quantitative evaluation of acoustic waves. Gradients of the wind field must be taken into account for precise quantification of ray trajectories and, consequently, for the localization of ground impacts, a key element for any micro‐barometric recording station.

Generalized paraxial ray tracing derived from Riemannian geometry

In 1973 R. White demonstrated that acoustic rays in a generic environment could be identified with the null geodesics of a pseudo-Riemannian manifold. A general set of paraxial (dynamic) ray equations, suitable for three dimensional ray tracing in a generic environment, is derived from the geodesic deviation equation used in general relativity and the geometric transmission loss of a ray bundle modeled from this equation. (When fluid motion is removed the paraxial ray procedure used in seismology emerges from the formalism.) The results are applied to time independent layered media where it is found that the standard ray integrals used in underwater acoustics as well as a generalized version of Snell’s law, originally derived by Kornhauser, emerge naturally as conserved quantities related to symmetries of the metric. Finally, when the results are applied to torsion free rays the deviation equation reduces to a scalar equation and the sectional curvature reduces to a simple expression depending on the derivatives of the sound speed and fluid velocity as well as the ray parameters allowing one to determine ray divergence or the development of focal points simply by checking the sign of a single term in the equation.

A paraxial schematic eye model for the growing C57BL/6 mouse

Purpose. The mouse eye has potential to become an important model for studies on the genetic control of eye growth and myopia. However, no data are published on the development of its optical properties. We developed a paraxial schematic model of the growing eye for the most common laboratory mouse strain, the C57BL/6 mouse, for the age range between 22 and 100 days. Methods. Refractive development was followed with eccentric infrared photorefraction and corneal curvature with infrared photokeratometry. To measure ocular dimensions, freshly excised eyes were immediately frozen after enucleation to minimize distortions. Eyes were cut with a cryostat down to the bisecting horizontal plane, until the optic nerve head became visible. The standard deviations were ±10 μm for repeated measurements in highly magnified videographs, taken in several section planes close to the equator in the same eyes. To evaluate inter-eye and inter-individual variability, a total of 20 mice (34 eyes) were studied, with 3–4 eyes for each of the 9 sampling ages. Schematic eye models were developed using paraxial ray tracing software (OSLO, LT Lambda Research Corporation, and a self-written program). Results. The measured refractive errors were initially +4.0 ± 0.6 D at approximately 30 days, and levelled off with +7.0 ± 2.5 D at about 70 days. Corneal radius of curvature did not change with age (1.414 ± 0.019 mm). Both axial lens diameter and axial eye length grew linearly (regression equations: lens, 1619 μm +5.5 μm/day, R=0.916; axial length, 2899 μm +4.4 μm/day, R=0.936). The lens grew so fast that vitreous chamber depth declined with age (regression equation: 896 μm −3.2 μm/day, R=0.685). The radii of curvature of the anterior lens surface increased during development (from 0.982 mm at day 22 to 1.208 mm at day 100), whereas the radii of the posterior lens surface remained constant (−1.081 ± 0.054 mm). The calculated homogeneous lens index increased linearly with age (from 1.568 to 1.605). The small eye artifact, calculated from the dioptric difference of the positions of the vitreo-retinal interface and the photoreceptor plane, increased from +35.2 to +39.1 D, which was much higher than the hyperopia measured with photorefraction. Retinal image magnification increased from 31 to 34 μm/deg, and the f/number remained ⩽1 at all ages, suggesting a bright retinal image. A calculated axial eye elongation of 5.4–6.5 μm was sufficient to make the schematic eye 1 D more myopic. Conclusions. The most striking features of the mouse eye were that linear growth was slow but extended far beyond sexual maturity, that the corneal curvature did not increase, and that the prominent lens growth caused a developmental decline of the vitreous chamber depth.

Difficult lens power calculations.

Although cataract extraction seems to be feasible without major technical obstacles, the surgical technique has changed completely, and patients are no longer satisfied with good spectacle-corrected vision but anticipate complete visual rehabilitation after cataract surgery, without correction. To fulfill this desire, toric or accommodative intraocular lenses are of increasing popularity, and the intraocular lens power calculation after keratorefractive surgery has been improved. In this review article, we provide an overview of different mathematical strategies of calculating the intraocular lens power with standard formulas and with new algorithms, such as paraxial or numeric ray-tracing. These enhanced techniques may improve the validity of lens power calculation due to reduction of the prediction error, especially in cases with high or excessive corneal astigmatism and after refractive laser surgery. Furthermore, a new calculation scheme for the determination of bitoric eikonic intraocular lenses allows a distortion-free imaging in astigmatic eyes. The biometric determinants for the different formulas and calculation schemes are discussed in detail. In difficult cases, standard calculation schemes are overemployed and new mathematical algorithms are necessary to adequately address these problems. Ray-tracing algorithms and other complex mathematical computation schemes are of increasing interest and will more and more replace conventional calculation formulas for determination of intraocular lens power.

Computerised calculation scheme for ocular magnification with the Zeiss telecentric fundus camera.

Littmann (1982) described a method to determine the magnification of the eye in order to relate the size of a retinal feature to its measured image size on a telecentric fundus camera film. This required information only about ametropia and corneal curvature. Several other methods have been reported since then which consider other biometric data to enhance the accuracy of this classical method. The purpose of this study is to describe a numerical calculation scheme to determine the magnification q of the eye in two cardinal meridians using paraxial raytracing. Our calculation scheme is based on ametropia, keratometry, as well as biometric data such as axial length, anterior chamber depth and thickness of the crystalline lens. It is described step-by-step in order (1) to determine the refractive powers of both surfaces of the crystalline lens, which are not directly measurable in vivo, (2) to derive the retinal image conjugate to a circular object using paraxial raytracing, (3) to fit an ellipse to the retinal image, (4) to determine the secondary principal points (Gaussian length) separately for both cardinal meridians and (5) to calculate the ocular magnification q. The power of the crystalline lens is estimated to compensate for the spherocylindrical refraction at the spectacle plane and the corneal refraction with an astigmatic component thus creating a sharp image focused at the retinal plane. The capabilities of this computing scheme are demonstrated with five clinical examples and are related to the respective values of the classical Littmann formula as well as to enhanced methods described by Bennett (1988), Bennett et al. (1994) and Garway-Heath et al. (1998).

Migration velocity analysis from locally coherent events in 2‐D laterally heterogeneous media, Part II: Applications on synthetic and real data

We demonstrate a method for estimating 2‐D velocity models from synthetic and real seismic reflection data in the framework of migration velocity analysis (MVA). No assumption is required on the reflector geometry or on the unknown background velocity field, provided that the data only contain primary reflections/diffractions. In the prestack depth‐migrated volume, locations where the reflectivity exhibits local coherency are automatically picked without interpretation in two panels: common image gathers (CIGs) and common offset gathers (COGs). They are characterized by both their positions and two slopes. The velocity is estimated by minimizing all slopes picked in the CIGs.We test the applicability of the method on a real data set, showing the possibility of an efficient inversion using (1) the migration of selected CIGs and COGs, (2) automatic picking on prior uncorrelated locally coherent events, (3) efficient computation of the gradient of the cost function via paraxial ray tracing from the picked events to the surface, and (4) a gradient‐type optimization algorithm for convergence.

Migration velocity analysis from locally coherent events in 2‐D laterally heterogeneous media, Part I: Theoretical aspects

We present a new method based on migration velocity analysis (MVA) to estimate 2‐D velocity models from seismic reflection data with no assumption on reflector geometry or the background velocity field. Classical approaches using picking on common image gathers (CIGs) must consider continuous events over the whole panel. This interpretive step may be difficult—particularly for applications on real data sets. We propose to overcome the limiting factor by considering locally coherent events. A locally coherent event can be defined whenever the imaged reflectivity locally shows lateral coherency at some location in the image cube.In the prestack depth‐migrated volume obtained for an a priori velocity model, locally coherent events are picked automatically, without interpretation, and are characterized by their positions and slopes (tangent to the event). Even a single locally coherent event has information on the unknown velocity model, carried by the value of the slope measured in the CIG. The velocity is estimated by minimizing these slopes.We first introduce the cost function and explain its physical meaning. The theoretical developments lead to two equivalent expressions of the cost function: one formulated in the depth‐migrated domain on locally coherent events in CIGs and the other in the time domain. We thus establish direct links between different methods devoted to velocity estimation: migration velocity analysis using locally coherent events and slope tomography.We finally explain how to compute the gradient of the cost function using paraxial ray tracing to update the velocity model. Our method provides smooth, inverted velocity models consistent with Kirchhoff‐type migration schemes and requires neither the introduction of interfaces nor the interpretation of continuous events. As for most automatic velocity analysis methods, careful preprocessing must be applied to remove coherent noise such as multiples.